Applications of Several Minimum Principles

Abstract

In our previous works, a Metatheorem in ordered fixed point theory showed that certain maximum principles can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore, there should be the dual principles related to minimality, anti-progressive maps, and others. In the present article, we derive several minimum principles particular to Metatheorem and their applications. One of such applications is the Brøndsted-Jachymski Principle. We show that known examples due to Zorn (1935), Kasahara (1976), Brézis-Browder (1976), Taskovi¢ (1989), Zhong (1997), Khamsi (2009), Cobzas (2011) and others can be improved and strengthened by our new minimum principles.

Keywords

• The 2023 Metatheorem
• Brøndsted-Jachymski Principle
• Zorn's Lemma
• Caristi fixed point theorem
• Ekeland variational principle
• preorder
• fixed point
• stationary point
• minimum principle

Kaynakça

1. [1] H. Brézis and F.E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976), 355-364.
2. [2] A. Brøndsted, Fixed point and partial orders, Shorter Notes, Proc. Amer. Math. Soc. 60 (1976), 365-366.
3. [3] S. Cobzas, Completeness in quasi-metric spaces and Ekeland variational principle, Topology Appl. 158 (2011), 1073-1084.
4. [4] S. Cobzas, Ekeland, Takahashi and Caristi principles in preordered quasi-metrc spaces, Quaestiones Mathematicae 2022: 1?22. https://doi.org/10.2989/16073606.2022.2042417
5. [5] I. Ekeland, Sur les problèmes variationnels, C.R. Acad. Sci. Paris 275 (1972), 1057?1059; 276 (1973), 1347-1348.
6. [6] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.
7. [7] J. Jachymski, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Analysis 52 (2003), 1455-1463.
8. [8] S. Kasahara, On fixed points in partially ordered sets and Kirk-Caristi theorem, Math. Seminar Notes 3 (1975), 229-232.

9. [9] M.A. Khamsi, Remarks on Caristi's fixed point theorem, Nonlinear Anal. 71(1-2) (2009), 227- 231.
10. [10] S. Park, Some applications of Ekeland's variational principle to fixed point theory, Approximation Theory and Applications (S.P. Singh, ed.), Pitman Res. Notes Math. 133 (1985), 159-172.
11. [11] S. Park, Countable compactness, l.s.c. functions, and fixed points, J. Korean Math. Soc. 23 (1986), 61-66.
12. [12] S. Park, Equivalent formulations of Ekeland's variational principle for approximate solutions of minimization problems and their applications, Operator Equations and Fixed Point Theorems (S.P. Singh et al., eds.), MSRI-Korea Publ. 1 (1986), 55-68.
13. [13] S. Park, Equivalents of various maximum principles, Results in Nonlinear Analysis 5(2) (2022), 169-174.
14. [14] S. Park, Applications of various maximum principles, J. Fixed Point Theory (2022), 2022-3, 1-23. ISSN:2052-5338.
15. [15] S. Park, Equivalents of maximum principles for several spaces, Top. Algebra Appl. 10 (2022), 68?76. 10.1515/taa-2022-0113.
16. [16] S. Park, Equivalents of generalized Brøndsted principle, J. Informatics Math. Sci., to appear.
17. [17] S. Park, Equivalents of ordered fixed point theorems of Kirk, Caristi, Nadler, Banach, and others, Adv. Th. Nonlinear Anal. Appl. 6(4) (2022), 420-432.
18. [18] S. Park, Extensions of ordered fixed point theorems, DOI: 10.13140/RG.2.2.21699.48160
19. [19] S. Park, Extensions of ordered fixed point theorems, II, to appear.
20. [20] S. Park, Applications of generalized Zorn's Lemma, DOI: 10.13140/RG.2.2.26690.04801
21. [21] S. Park, Generalizations of the Tarski type fixed point theorems, to appear.
22. [22] S. Park, Foundations of ordered fixed point theory, J. Nat. Acad. Sci., ROK, Nat. Sci. Ser. 61(2) (2022).
23. [23] M.R. Taskovi¢, On an equivalent of the axiom of choice and its applications, Math. Jpn. 31(6) (1986), 979-991.
24. [24] C.-K. Zhong, On Ekeland's variational principle and a minimax theorem, J. Math. Anal. Appl. 205 (1997), 239-250.